Correction

After the publication of this work [1], we discovered programming errors in our software implementation of the proposed
error-weighted, uncorrelated shrunken centroid (EWUSC) algorithm and the uncorrelated
shrunken centroid (USC) algorithm. We have corrected these errors, and the updated
results are summarized in the revised Table 6.

On the NCI 60 data, both Figure 1 in [1] and the revised Figure 1 showed that USC generally produces higher prediction accuracy than the 'shrunken
centroid' algorithm (SC) [2] using the same number of relevant genes. Using the revised software implementation,
USC requires fewer (2,116 instead of 2,315 as reported in [1]) genes to achieve 72% accuracy. The number of genes required by SC to achieve the
same prediction accuracy remains the same (3,998).

Figure 1. A corrected figure showing the comparison of prediction accuracy of USC and SC on
the NCI 60 data. The percentage of prediction accuracy is plotted against the number
of relevant genes using the USC algorithm at ρ0 = 0.6 and the SC algorithm (USC at ρ0 = 1.0). The horizontal axis is shown on a log scale. Because no independent test set
is available for this data, we randomly divided the samples in each class into roughly
three parts multiple times, such that a third of the samples are reserved as a test
set. Thus the training set consists of 43 samples and the test set of 18 samples.
The graph represents typical results over these multiple random runs.

Figure 2 shows the results of applying EWUSC to the training set, four-fold cross-validation
data, and test set of the multiple tumor data over a range of shrinkage thresholds
(Δ) and correlation thresholds (ρ0). The revised Figure 2 shows the same general trend as Figure 2 in [1]: the percentage of errors is reduced when ρ0 < 1 over most values of Δ on the training set, cross-validation data and test set;
Figure 2d shows that the number of relevant genes is drastically reduced when genes with correlation
threshold above 0.9 are removed. The values of the optimal shrinkage thresholds (Δ)
determined from the cross-validation results have changed using the revised implementation.
Specifically, the optimal shrinkage threshold values (Δ) for both EWUSC and USC are
reduced to 4.8 and 4 respectively (see revised Table 6). The numbers of relevant genes selected by EWUSC and USC are reduced and the resulting
prediction accuracy for both USC and SC are also reduced in the revised results. In
the case of using the global optimal parameters when Δ = 0, the EWUSC in the revised
implementation selected slightly fewer genes (1,622 instead of 1,626) at the expense
of slightly lower prediction accuracy (74% instead of 78%). Figure 4 compares the
prediction accuracy on the test set of the multiple tumor data using the EWUSC and
USC algorithms at the estimated optimal correlation threshold (ρ0 = 0.8), the SC algorithm and the Support Vector Machine (SVM). The general observations
previously reported in [1] still hold with the revised Figure 4. First, USC produces higher prediction accuracy than SC using the same number of
relevant genes. Second, EWUSC generally produces higher prediction accuracy than USC
using the same number of relevant genes. In fact, the performance of EWUSC is stronger
than previously reported in [1] when the number of genes is small.

Figure 2. A corrected figure showing the prediction accuracy on the multiple tumor data using
the EWUSC algorithm over the range of Δ from 0 to 20. The percentage of classification
errors is plotted against Δ on (a) the full training set (96 samples) and (c) the test set (27 samples). In (b) the average percentage of errors is plotted against Δ on the cross-validation data
over five random runs of fourfold cross-validation. In (d), the number of relevant genes is plotted against Δ. Different colors are used to
specify different correlation thresholds (ρ0 = 0.6, 0.7, 0.8, 0.9 or 1). Optimal parameters are inferred from the cross-validation
data in (b).

Figure 4. A corrected figure showing the comparison of prediction accuracy of EWUSC (ρ0 = 0.8), USC (ρ0 = 0.8), SVM and SC algorithms on the multiple tumor data. The horizontal axis shows
the total number of distinct genes selected over all binary SVM classifiers on a log
scale. Some results are not available on the full range of the total number of genes.
For example, the maximum numbers of selected genes for EWUSC and USC are roughly 1,000.
The reported prediction accuracy is 78% [5] using all 16,000 available genes on the
full data. The EWUSC algorithm achieves 85% prediction accuracy with only 77 genes.
With 241 genes, EWUSC produces 93% prediction accuracy.

Figure 5 shows the comparison of prediction accuracy of EWUSC, USC, and SC on the breast cancer
data. With the revised implementation, the optimal correlation threshold (ρ0) is changed from 0.7 in [1] to 0.6 (see revised Table 6). The observation reported in [1] that EWUSC produces higher prediction accuracy on the test set than USC and SC when
the number of relevant genes is small still holds. The numbers of relevant genes selected
by USC and SC are significantly larger with the revised implementation (see revised
Table 6).

Figure 5. A corrected figure showing the comparison of prediction accuracy of EWUSC, USC and
SC on the breast cancer data. The percentage of prediction accuracy is plotted against
the number of relevant genes using the EWUSC algorithm at ρ0 = 0.6, the USC algorithm at ρ0 = 0.6 and the SC algorithm (USC at ρ0 = 1.0). Note that the horizontal axis is shown on a log scale.

The major conclusions and observations in the original manuscript [1] remain valid with the revised implementation. Our EWUSC and USC algorithms represent
improvements over the SC algorithm. In general, fewer genes are required to produce
comparable prediction accuracy. On the multiple tumor data, our EWUSC and USC algorithms
produce higher prediction accuracy using fewer relevant genes compared to published
results. The revised software implementation is available on our web site [3]. Note: the revised version (1.0) of the software was placed on the web site on May
9, 2005.